Overview
This research introduces an extension of the Berestycki–Monneau–Scheinkman (BMS) model, a framework used for analyzing speculative financial bubbles. The primary modification involves allowing the investor disagreement process, which is a core component of the BMS model, to exhibit jumps rather than following continuous paths. This extension incorporates an independent Lévy jump process to account for sudden, discontinuous shifts in market sentiment.
The resulting model characterizes the speculative bubble premium through a non-local partial integro-differential equation (PIDE) that includes a moving obstacle. The study develops a viscosity solution theory specifically for this non-local obstacle problem. It establishes a comparison principle and demonstrates the existence and uniqueness of the bubble price by employing Perron's method, which involves constructing explicit continuous sub- and supersolutions.
Research Context
The original BMS framework for speculative financial bubbles posits that investor disagreement evolves continuously. The present work departs from this assumption by recognizing that market sentiment can experience sudden, rather than gradual, changes. To capture these discontinuous dynamics, the model integrates a Lévy jump process into the investor disagreement component, thereby extending the scope of the BMS model to scenarios involving abrupt market shifts.
Approach
The methodology employed in this study combines several theoretical and computational techniques:
- Model Formulation: The extended model incorporates an independent Lévy jump process to describe sudden shifts in market sentiment within the investor disagreement process. This contrasts with the continuous path assumption of the original BMS framework.
- Mathematical Framework: Using optimal stopping theory and the Itô–Lévy formula, the speculative bubble premium is shown to satisfy a non-local partial integro-differential equation (PIDE) that includes a moving obstacle.
- Viscosity Solution Theory: A viscosity solution theory is developed for the derived non-local obstacle problem. This theoretical framework provides a means to analyze solutions to the PIDE.
- Existence and Uniqueness Proofs: A comparison principle is proven via a doubling-of-variables argument, specifically adapted to handle the non-local jump integral. Existence and uniqueness of the bubble price are then established using Perron's method, which involves constructing explicit continuous sub- and supersolutions.
- Numerical Scheme Development: A monotone Implicit–Explicit finite difference scheme is introduced for the calculation of the bubble premium.
- Scheme Convergence: Following the Barles–Souganidis framework, it is demonstrated that the discrete operator within the scheme preserves the M-matrix property. The scheme's convergence locally uniformly to the unique viscosity solution is shown under a state-dependent Courant–Friedrichs–Lewy (CFL) condition.
- Implementation and Testing: The developed scheme is implemented through a PSOR–Picard algorithm. Numerical tests are presented for four distinct Lévy models, encompassing both finite-activity and infinite-activity characteristics.
Findings
The study yielded several key findings:
- The extension of the BMS model to include an independent Lévy jump process for investor disagreement results in a speculative bubble premium that satisfies a non-local partial integro-differential equation (PIDE) with a moving obstacle.
- A viscosity solution theory was successfully developed for this specific non-local obstacle problem.
- A comparison principle was proven using a doubling-of-variables argument tailored to the non-local jump integral.
- Existence and uniqueness of the bubble price were established through Perron's method, supported by the construction of explicit continuous sub- and supersolutions.
- A monotone Implicit–Explicit finite difference scheme for the bubble premium was introduced and shown to preserve the M-matrix property of the discrete operator, following the Barles–Souganidis framework.
- The scheme converges locally uniformly to the unique viscosity solution, provided a state-dependent Courant–Friedrichs–Lewy (CFL) condition is met.
- Numerical tests were conducted using a PSOR–Picard algorithm for four different Lévy models, including both finite- and infinite-activity types.
Why This Matters
This research offers a more generalized mathematical framework for modeling financial bubbles by explicitly incorporating the possibility of sudden, discontinuous shifts in market sentiment, a phenomenon not captured by models assuming continuous sentiment evolution. By providing a rigorous theoretical foundation, including existence and uniqueness proofs for the bubble price in this extended setting, and developing a convergent numerical scheme, the work provides tools for analyzing and potentially better understanding the dynamics of speculative bubbles in markets where abrupt changes are a factor. The numerical tests performed with various Lévy models demonstrate the applicability of the developed methods to different types of discontinuous market behavior.